Conditional measures of deterninantal point processes: the Gibbs property and the Lyons–Peres conjecture

Determinantal point processes arise in many different problems: spanning trees and Gaussian zeros, random matrices and representations of infinite-dimensional groups. How does the determinantal property behave under conditioning?
The talk will first address this question for specific examples such as the sine-process, where one can explicitly write the analogue of the Gibbs condition in our situation.
We will then consider the general case, where, in joint work with Yanqi Qiu and Alexander Shamov, proof is given of the Lyons–Peres conjecture on completeness of random kernels.
The talk is based on the preprint arXiv:1605.01400 as well as on the preprint arXiv:1612.06751 joint with Yanqi Qiu and Alexander Shamov.